Mechanism analysis of a trisector1

《Mechanism analysis of a trisector1》由会员分享，可在线阅读，更多相关《Mechanism analysis of a trisector1（3页珍藏版）》请在装配图网上搜索。

1、Mechanism analysis of a trisectorLyndon O. Barton*Delaware State University, Dover, Delaware, United StatesAbstractThis paper presents a graphical procedure for analyzing a trisector - a mechanism used for trisecting an arbitrary acute angle. The trisector employed was a working model designed and b

2、uilt for this purpose. The procedure, when applied to the mechanism at the 60° angle, which has been proven to be not trisectable as well as the 45° angle for benchmarking (since this angle is known to be trisectable), produced results that compared remarkably in both precision and accurac

3、y.For example, in both cases, the trisection angles found were 20.00000°, and 15.00000°, respectively, as determined by The Geometers Sketch Pad software. Considering the degree of accuracy of these results (i.e. five decimal places) and the fact that it represents the highest level of pre

4、cision attainable by the software, it is felt that the achievement is noteworthy, notwithstanding the theoretical proofs of Wantzel, Dudley, and others Underwood Dudley, A Budget of Trisections, Springer-Verlag, New York, 1987; Clarence E. Hall, The equilateral triangle, Engineering Design Graphics

5、Journal 57 (2) (1993); Howard Eves, An Introduction to The History of Mathematics, sixth ed., Saunders College Publishing, Fort Worth, 1990; Henrich Tietze, Famous Problems of Mathematics, Graylock Press, New York, 1965.Keywords: Mechanism analysis; Angle trisector; Four bar mechanism; Slider crank

6、mechanism; Slider-coupler mechanism; Famous problems in mathematics1. IntroductionThe problem of the trisection of an angle has been for centuries one of the most intriguing geometric challenges for mathematicians 1-4. According to Underwood Dudley 1 author of A Budget of Trisections,Certain angles

7、can be trisected without difficulty. For example, a right angle can be trisected, since an angle of 30° can be constructed. However, there is no procedure, using only an unmarked straight edge and compasses, to construct one-third of an arbitrary angle.Dudley then proceeded to lay out a proof o

8、f this statement by showing that a 60° angle cannot be trisected. Also, in the same text, he referenced the work of Pierre Laurent, Wantzel, who in 1837 first proved that such trisection was impossible. Yet in a more recent paper by Hall 2, a proof was presented to show that a three to. one rel

9、ationship between certain angles can exist for acute angles. However, Hall 2 did not develop or present a procedure for the trisection.The purpose of this paper is not to contradict or debate the established proofs alluded to, but to present a summary of the results obtained from my study of a trise

10、ctor mechanism, which I designed and built as part of the study 5. It is hoped that these results as well as the approach used in developing them will provide others in the mathematics and science community valuable physical insights into the nature of the problem.The results presented are based on

11、an analysis, using unmarked ruler and compasses only (aided by The Geometers Sketch Pad software), of the 60° angle that has been proven to be not trisectable as well as the 45° angle that is known to be trisectable.2. Theory The proposed method is based on the general theorem relating arc

12、s and angles.Let ZECG (or 30) be the required angle to be trisected. With center at C and radius CE describe a circle. Given that a line from point E can be drawn to cut the circle at S and intersect the extended side GC at some point M such that the distance SM is equal to the radius SC, then from

13、the general theorem relating to arcs and angles, 3. Trisector design and analysisThe trisector mechanism illustrated in Fig. 1 is a compound mechanism consisting of a four-bar linkage, CEDA, where CE is the crank, ED is the coupler, and DA is the follower, and a slider-crank linkage 6, CFVE", w

14、here CV is the crank, FE" the connecting rod, and F the slider. The links for the four-bar and slider-crank are designed so that the pin joints are all located at equal distances apart, and both linkages are mounted on a common base and connected at fixed axis C, where the two cranks CE and CV

15、meet, as well as at crank pin E, via a pivoting slot through which the connecting rod slides. Thus, as the four-bar crank CE is rotated in one direction or the other, between the 90° and 0° positions, the connecting rod FE" of the slider-crank divides the angle formed by said crank an

16、d coupler into a 2-1 ratio. Also, if crank CE is rotated, say in a clockwise manner, the connecting rod FE" would be forced to undergo combined motion of translation, where sliding would occur at both ends, and rotation only at the pivoting slot end. Meanwhile crank CV of the slider-crank would

17、 be forced to rotate, but in a counterclockwise manner.By assuming slider F is not constrained (or not restricted by the slot), while other parts of the mechanism are in motion, the mechanism behaves like a sliding coupler mechanism 6 where, it was possible to show that the path point T or joint at

18、slider F is practically a smooth circular path (See Appendix Fig. A1). This path intercepts the straight path that F would normally describe, when constrained within the slot, at a unique point. Further, the analysis will show that this point locates the vertex of the angle formed by the connecting

19、rod and said slot, which represents the required trisection angle.4. Procedure for Figs. 1A or 2AReferring to Figs. 1A and 2A let CEDA and CVFE" represent the four bar and slider crank components, respectively of the trisector, as shown in the 90° position 1. With center at C and radius CE

20、, construct an arc from point E to point G on the ground to represent the path of E as crank CE of the four-bar is rotated between its normal position at 90° and the ground at 0°.2. Draw the four-bar in its 60° or 45° position where ZE'CG is equal to 60° or 45°.3. A

21、ssuming point F to be a fixed axis, temporarily, join points F and E' with a segment FE' to represent the connecting rod segment FH" when rotated about F to its new position dictated by crank CE'.4. Still assuming point F to be fixed, but crank CV disconnected from the connecting ro

22、d, construct an arc cutting FE' at W. The arc VW then represents the path of V as the rod segment FH is rotated to FE'.5. Now, assume that F is not fixed, and also not constrained by the fixed slot, then the rod FE" will change position to become TE', while crank CV will be rotated

23、accordingly to a new position. Therefore,(a) extend segment FE' from point F to point T to represent HE" (i.e. the portion of the connecting rod that slides within the pivoting slot at E),(b) with center at C and radius CV, describe an arc from V cutting TE' at point V. The arc VV"

24、 will represent the path of V between its original position on FE" to its final position on TE'.Note that in the new configuration of the linkage, because F is assumed to be unconstrained or free to move, point T falls below the ground, and TE' represents the new position of the connect

25、ing rod FE". Therefore TE" = HE" = V'W, where V'W represents the change in positions of point V along the relocated connecting rod TE'.5. Procedure for Figs. 1B or 2B1. With the linkage in any given acute angular position (for example, 60° or 45° being considered

26、 in this paper), join points A and V with a segment AV.2. From point T, construct a line TX parallel to AV and from the same point another line TY perpendicular to ground AC.3. Bisect the angle formed by lines TX and TY above and define the point where the bisector intersects ground AC as point P.4.

27、 Connect point E' to point P with segment E'P and define intersection of this segment with the circular path AV as point R.With center at R and radius RC, construct an arc to cut RP at N, where N defines the unconstrained end of the connecting rod in some intermediate position. 6. Procedure

28、for Figs. 1C or 2CReferring to Figs. 1C or 2C, join points N and T with segment NT and construct a bisector of this segment to cut link extension E0C at point O, thereby establishing the center of the circular path of the unconstrained end of the connecting rod. With center at O and radius TO, const

29、ruct an arc to intersect ground AC and define the point of intersection as point M. This point locates the vertex of the angle formed by the connecting rod and the ground AC.Join points E0 and M to form segment E0M and angle E0MA, which represents the required trisection angle (note that E0M cuts ar

30、c AV at S)7. SummaryA graphical procedure for analyzing a trisector mechanism has been presented. The procedure, when applied to the mechanism at the 60° angle, which has been proven to be not trisectable, as well as the 45° angle, which is known to be trisectable, has yielded results that

31、 compared remarkably in both precision and accuracy.For example, the results, as determined by The Geometers Sketch Pad software, for the angles 60° and 45° were, respectively 20.00000° and 15.00000° (i.e. five decimal places) which represented the highest level of precision attainable by the software. Thus, it is felt that this degree of accuracy is not only significant, but noteworthy, notwithstanding the theoretical proofs by Wantzel, Dudley, and others 1-4.